Chapter 3 Motion in Two and Three Dimensions - Chapter3...

Chapter 3 Motion in Two and Three Dimensions 3.1 The Important Stuff 3.1.1 Position In three dimensions, the location of a particle is specified by its location vector, r: r = xi + yj + zk (3.1) If during a time interval t the position vector of the particle changes from r1 to r2, the displacement r for that time interval is r = r1 ' r2 (3.2) = (x2 ' x1)i + (y2 ' y1)j + (z2 ' z1)k (3.3) 3.1.2 Velocity If a particle moves through a displacement r in a time interval t then its average velocity for that interval is v = r t =xti + ytj + zt k (3.4) As before, a more interesting quantity is the instantaneous velocity v, which is the limit of the average velocity when we shrink the time interval t to zero. It is the time derivative of the position vector r: v = dr dt (3.5) = d dt (xi + yj + zk) (3.6) = dx dt i + dy dt j + dz dt k (3.7) can be written: v = vxi + vyj + vzk (3.8) 51 52 CHAPTER 3. MOTION IN TWO AND THREE DIMENSIONS where vx =

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dx dt vy = dy dt vz = dz dt (3.9) The instantaneous velocity v of a particle is always tangent to the path of the particle. 3.1.3 Acceleration If a particles velocity changes by v in a time period t, the average acceleration a for that period is a = vt = vx ti + vy tj + vz t k (3.10) but a much more interesting quantity is the result of shrinking the period t to zero, which gives us the instantaneous acceleration, a. It is the time derivative of the velocity vector v: a = dv dt (3.11) = d dt (vxi + vyj + vzk) (3.12) = dvx dt i + dvy dt j + dvz dt k (3.13) which can be written: a = axi + ayj + azk (3.14) where ax =

dvx dt = d2x dt2 ay = dvy dt = d2y dt2 az = dvz dt = d2z dt2 (3.15) 3.1.4 Constant Acceleration in Two Dimensions When the acceleration a (for motion in two dimensions) is constant we have two sets of equations to describe the x and y coordinates, each of which is similar to the equations in Chapter 2. (Eqs. 2.62.9.) In the following, motion of the particle begins at t = 0; the initial position of the particle is given by r0 = x0i + y0j and its initial velocity is given by v0 = v0xi + v0yj and the vector a = axi + ayj is constant. vx = v0x + axt vy = v0y + ayt (3.16) x = x0 + v0xt + 1 2axt2 y = y0 + v0yt + 1 2ayt2 (3.17) v2 x = v2 0x + 2ax(x ' x0) v2 y = v2 0y + 2ay(y ' y0) (3.18) x = x0 + 1 2 (v0x + vx)t y = y0 + 1 2 (v0y + vy)t (3.19) Though the equations in each pair have the same form they are not identical because the components of r0, v0 and a are not the same. 3.1. THE IMPORTANT STUFF 53

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